Representing real numbers on the number line

Correct answer: A. 4 units

The distance is the absolute difference: |−2 − (−6)| = |4| = 4 units. Since −2 is to the right of −6, you move 4 units to the right.

Correct answer: A. -2.5

Moving to the left on a number line means subtracting. Therefore, the required point is \(0-2.5=-2.5\). Option B represents moving 2.5 units to the right instead. Exam tip: Numbers to the left of 0 on the number line are negative.

Correct answer: A. 2

Moving right increases the number, so add 3 to the starting point: −1 + 3 = 2. The point reached is 2.

Correct answer: A. Both are equal

To compare a decimal with a fraction, convert one representation into the other. The fraction 3/2 means three divided by two, and 3 ÷ 2 = 1.5. Equivalently, writing 1.5 as a fraction gives 15/10, which reduces by dividing numerator and denominator by 5 to 3/2. Since the two values are exactly equal, they correspond to the same point on the number line, not merely to nearby points. Therefore option A is correct. Option B and option C would require one value to be smaller, but equality has already been established. Option D is also false because opposite numbers have equal magnitudes with opposite signs, such as 1.5 and −1.5; both numbers here are positive and equal. The governing concept is equivalence of decimal and fractional forms.

Correct answer: A. (\frac{1}{3})

(0.333\ldots=\frac{1}{3}) is a recurring decimal. Recurring decimals also represent fixed points on the number line.

Correct answer: A. (1)

(0.999\ldots=1), so both are the same point. Do not treat a recurring decimal like a terminating decimal.

Correct answer: A. (2) and (3)

Since (2^2=4) and (3^2=9), (\sqrt{7}) lies between (2) and (3). Bound the number using nearby squares.

Correct answer: A. (-3) and (-2)

(\sqrt{8}) lies between (2) and (3), so (-\sqrt{8}) lies between (-3) and (-2). The negative sign changes the direction.

Correct answer: A. 2.10

The number 2.10 can also be written as 2.1. Comparing the decimals by place value, the tenths digit of 2.10 is 1, while that of 2.01 is 0. Hence, 2.10 is greater and lies farther to the right on the number line. Exam tip: Compare the whole-number part first, followed by the tenths, hundredths, and later places.

Correct answer: A. (\frac{1}{2})

(\frac{2}{5}=0.4), (\frac{1}{2}=0.5), and (\frac{3}{5}=0.6), so (\frac{1}{2}) lies between them. Decimal form helps in comparison.

Correct answer: A. -3

\(\frac{-6}{2}=-3\) because 6 divided by 2 is 3, and the result remains negative since the numerator is negative. Therefore, the number is represented by the point -3 on the number line. Option 3 ignores the negative sign. Exam tip: simplify the fraction and check its sign before locating the point.

Correct answer: A. 2

A fraction represents division of its numerator by its denominator. Therefore 8/4 means 8 divided by 4, which equals 2 because four fits into eight exactly two times. The fraction can also be simplified by dividing numerator and denominator by their common factor 4: 8/4 = (8÷4)/(4÷4) = 2/1 = 2. Hence the point 8/4 coincides with the integer 2 on the number line, so option A is correct. Option B confuses the denominator with the value of the fraction; option C repeats the numerator rather than performing the division; and option D would represent 1, which is obtained only when numerator and denominator are equal. The governing concept is fraction simplification and the identification of equivalent numerical values.

Correct answer: A. left side including (2)

(x\le 2) means (2) or smaller numbers, so it is the left side including (2). The symbol (\le) includes the boundary point.

Correct answer: A. right side excluding (-1)

(x>-1) means numbers greater than (-1), so the region is to the right and (-1) is not included. The symbol (>) gives an open point.

Correct answer: A. 2.5

The number exactly midway between two numbers is their average. Thus, \(\frac{1+4}{2}=\frac{5}{2}=2.5\). Hence, 2.5 is correct because it is 1.5 units from both 1 and 4. The number 3 is closer to 4, while 2 is closer to 1. Exam tip: Find the midpoint of \(a\) and \(b\) using \(\frac{a+b}{2}\).

Correct answer: A. (3) and (4)

Since (3^2=9) and (4^2=16), (\sqrt{13}) lies between (3) and (4). Use nearby perfect squares to locate square roots.

Correct answer: A. (-3) and (-2)

(-\frac{9}{4}=-2.25), so it lies between (-3) and (-2). Be careful with direction for negative fractions.

Correct answer: A. 1/2

The exact midpoint of two numbers is their arithmetic mean, found by adding the endpoints and dividing by 2. For −2 and 3, the calculation is (−2 + 3)/2 = 1/2. Thus 1/2 is the number equally distant from both endpoints: the distance from −2 to 1/2 is 2.5 units, and the distance from 1/2 to 3 is also 2.5 units. Therefore option A is correct. Option B is one unit from 0 but is not equally distant from the given endpoints; its distances are 3 and 2. Option C lies on the negative side and is not the average, while option D is outside the interval altogether. The governing concept is the midpoint formula (a+b)/2, which works for positive, negative, and mixed endpoints alike.

Correct answer: A. Between (0) and (1)

(\frac{3}{4}) is greater than (0) and less than (1). In exams first compare the fraction value with nearby integers.

Correct answer: A. −3 and −2

The governing concept is the order of integers on a number line. A number lies between two consecutive integers when it is greater than the left integer and less than the right integer. For −2.5, the correct comparison is −3 < −2.5 < −2. Thus −2.5 lies between −3 and −2, exactly halfway between them. Negative numbers require careful attention because the integer with the larger absolute value is farther left: −3 is left of −2. Option B describes the interval immediately to the right of −2, option C contains positive numbers, and option D is the interval to the left of −3. Therefore, option A is the only interval containing −2.5.

Correct answer: B. \(3\)

\(\sqrt{9}\) means the principal (non-negative) square root of \(9\). Since \(3^2=9\) and square roots are non-negative, \(\sqrt{9}=3\). So the point is \(3\).

Correct answer: B. (2) and (3)

Since (2^2<5<3^2), (\sqrt{5}) lies between (2) and (3). Use squares to locate square roots quickly.

Correct answer: C. (1.2)

Moving right from (0) gives a positive number, so (A=1.2). Check direction first to decide the sign.

Correct answer: A. (-0.5)

(-\frac{1}{2}=-0.5), so both represent the same point. Converting fractions to decimals makes comparison easier.

Correct answer: C. Third part after (1)

(1.75=1+\frac{3}{4}), so it lies at the third equal part after (1). Converting decimals to fractions helps locate points easily.